>>12417722No shit. My whole point is that sqrt(2) is undefined unless you specify what ring you're talking about.
>Didn't say that it did. I said that Q[X]/(X2?2) does.Of course, that's completely obvious. My point is that your definition concerns an arbitrary ring and general rings don't have an element x s.t. x*x=1+1.
>>12418284> Are you the strokes on a board guy?Yeah.
>>12418309Sorry I made a typo for multiplication there.
You're right to be concerned.
I meant to say
(a,b)*(c,d)=(ac+2bd, ad+bc).
Then it works out that (0,1)*(0,1)=(2,0)
And the arithmetic works just like you would expect when you identify (a,b) with a+bsqrt(2).
>>12418325Based.
>>12418331This is a nice attempt, to define numbers in terms of lengths of line segments in Euclidean geometric constructions. Your definition doesn't agree at all with the modern mathematical definition of what it is (usually an equivalence class of Cauchy sequences or a Dedekind cut).
However, at most you can extend this notion of "length" to constructible numbers, which is a strict subfield of the real number. So if you really want to go with that, you no longer have the numbers pi, e, or whatever, unless you arithmetize your geometry with real numbers, in which case you can no longer define numbers as length but rather will need to go back to the classical definitions of the reals.
>>12418439Of course in practice it is, but it's a very bad abuse of notation, because in order for it to be consistent, equality is no longer transitive, which means it has no right to be called an equality. If you really want a shorthand (which is actually even shorter) for what you wrote, just write
.
